Ulam-Hyers stability of tuberculosis and COVID-19 co-infection model under Atangana-Baleanu fractal-fractional operator

The intention of this work is to study a mathematical model for fractal-fractional tuberculosis and COVID-19 co-infection under the Atangana-Baleanu fractal-fractional operator. Firstly, we formulate the tuberculosis and COVID-19 co-infection model by considering the tuberculosis recovery individuals, the COVID-19 recovery individuals, and both disease recovery compartment in the proposed model. The fixed point approach is utilized to explore the existence and uniqueness of the solution in the suggested model. The stability analysis related to solve the Ulam-Hyers stability is also investigated. This paper is based on Lagrange’s interpolation polynomial in the numerical scheme, which is validated through a specific case with a comparative numerical analysis for different values of the fractional and fractal orders.


Basic definitions
In this segment, we will discuss some basic concepts related to the fractal-fractional operator and some known definitions that will be needed to obtain the main results of this study. Also, in this work, we assume the space {y(s) ∈ C([0, 1]) → R} with �y� = max s∈[0,1] |y(s)|.

Definition 1
Let y ∈ C((a, b), R) be a fractal differentiable on (a, b). The fractal-fractional derivative of y(s) with fractional order 0 < k 1 ≤ 1 and fractal dimension 0 < k 2 ≤ 1 in the sense of Atangana-Baleanu having a generalized Mittag-Leffler type kernel can be defined as follows 33 : where A B (k 1 ) = 1 − k 1 + k 1 Ŵ(k 1 ) and dy(u) du k 2 = lim s→u y(s)−y(u) Definition 2 For the same function y, considered above, the fractal-fractional integral of y(s) with fractional order 0 < k 1 ≤ 1 in the sense of Atangana-Baleanu having a Mittag-Leffler type kernel can be defined as follows 31 : Ethical approval. This article does not contain any studies with human participants or animals performed by any of the authors.

Model formulation
This segment describes a TB and COVID-19 co-infection model based on the Atangana-Baleanu fractal-fractional operator. Our model given below is an extension of some specified in 17,18 by, www.nature.com/scientificreports/ • Including the COVID-19 disease reinfection of recovered individuals; and • Including the TB recovery compartment, the COVID-19 recovery compartment and both diseases recovery compartments; and • Including the COVID-19 infection after recovery from TB and TB infection after recovery from  Under the schematic diagram given in Fig. 1, the TB and COVID-19 co-infection model is presented by the system of equations depicted as follows: where and N(s) = S TC + L T + I T + R T + I RC + A C + I C + R C + I RT  (1)   (1), the human population is divided into twelve compartments: Susceptible to both TB and COVID-19 (S TC ) , latent level TB infected people (L T ) , active level TB infected people (I T ) , recovered from TB (R T ) , COVID-19 infection after recovery from TB (I RC ) , COVID-19 infected with asymptomatic (A C ) , COVID-19 infected with symptomatic (I C ) , recovered from COVID-19 (R C ) , TB infection after recovery from COVID-19 (I RT ) , latent TB and COVID-19 dual infected compartment (L TC ) , TB and COVID-19 dual infected compartment (I TC ) , recovered people from both diseases (R). Table 1 describes the suggested model parameters.
We assumed that the susceptible people had been recruited into the constant rate π and the susceptible class develops TB through contact with active level TB infected patients by a force of infection T , expressed as This expression says that 1 represents the transmission rate of TB infection. The latent TB infection is considered asymptomatic and does not spread the disease. Similarly, susceptible people acquire infection with COVID-19 following effective contact with people infected with COVID-19 at a force of infection for COVID-19 C , given as here 2 denotes the COVID-19 disease transmission rate. Furthermore, we considered the individuals in the latent level TB infected people compartment ( L T ) leave to active level TB infected people compartment ( I T ) at a rate of latent TB infected people to become infected α 1 , and to both diseases latent infection compartment at a force of infection η T and some component is the rate of recovered from latent TB infected people ω 1 . Additionally, individuals with the TB disease infection ( I T ) after recovering from active TB at a rate of ρ 1 while the remaining component shifted to both diseases infection ( I TC ) at both diseases infectious rate of θ 1 or TB infected people die www.nature.com/scientificreports/ due to the death rate of d T . The recovered from TB ( R T ) has left either compartment ( I RC ), ν 1 is respectively, the rate of COVID-19 infection after recovery from TB. Then both diseases infected in latent level ( I RC ) move to the compartment (R) at a rate of both diseases recovered. Moreover, we considered the individuals in the asymptomatic COVID-19 compartment ( A C ) leave to infected COVID-19 compartment ( I C ) at a rate of asymptomatic COVID-19 infected people α 2 , and to both diseases infection a force of infection ǫ T and some component is the recovery rate of asymptomatic COVID-19 infected people ω 2 . Similarly, the individuals of the COVID-19 disease infection ( I C ) become recovered from COVID-19 at a rate of ρ 2 or shifted to both diseases infection ( I TC ) and both diseases are infectious at a rate of θ 2 and d C respectively, COVID-19 disease death rate in this compartment. In addition, the recovered from COVID-19 ( R C ) has the chance to leave either compartment ( I RT ), respectively, at a rate of ν 2 . Then both latent COVID-19 and TB coinfected individuals ( I RT ) move to the compartment (R) at a recovery rate from COVID-19 and TB sequentially. The latent co-infection diseases population in the compartment ( L TC ) either progresses to the co-infection ( L TC ) at a rate α 12 . The remaining component is assumed to be shifted to either compartment at a σ as illustrated in Fig. 1. That is, the susceptible people in the compartment ( L TC ) move to ( I T ) with a rate of recovery in COVID-19 people p σ 2 , move to the I C with a rate of recovery in TB infected people of p σ 1 , and become recovered at a rate of (1 − (p 1 + p 2 ))σ . Moreover, we considered that both diseases dual infection I TC leave compartments ( I T , I C , and R) denoted at a rate of m 1 τ , m 2 τ , or (1 − (m 1 + m 2 ))τ while the co-infection induced death rate is d TC . Finally, recovered from both TB and COVID-19 (R) at the rate of natural death is denoted by µ.

Existence and uniqueness results
In this segment, we utilize the fixed-point procedure to present the existence and uniqueness of the solution for the proposed model. Applying the Atangana-Baleanu fractal-fractional integral operator on the model (1) and utilizing the initial conditions, we obtain

Theorem 1
The Lipschitz condition is satisfy the Q i for i ∈ N 12 1 , if the assumption H is holds true and fulfills and � i < 1 , for i ∈ N 12 1 .
Proof Now, we prove that Q 1 (s, S TC ) fulfills the Lipschitz condition. For S TC (s) , S TC (s) , we get Hence, all the kernels Q i , i ∈ N 12 1 satisfies the Lipschitz property with constant � i < 1 for i ∈ N 12 1 . The proof is completed. Now, Eqs. (2) to (13) can be rewrite as follows: Proof We define the functions as follows: Then, we find that

Ulam-Hyers stability of the proposed problem
In this segment, we obtain the Ulam-Hyers stability of the proposed model (1). We state the required definition.

Numerical scheme
In this segment, the numerical scheme are analyzes for the proposed model (1). For the numerical scheme, we consider the equation of the Atangana-Baleanu fractional operator can also as follows: Utilizing the fractal-fractional integral operator having generalized Mittag-Leffler type kernel, we obtain Now, at s = s n+1 , which gives which can be written as Utilizing the Lagrange polynomial interpolation to Eq. (39), we obtain For clarity, we can write the as follows: Thus, by assuming .
s k 2 −1 γ Q 5 (s γ , I RC (s γ ))y 1 (n, γ ) − s k 2 −1 γ −1 Q 5 (s γ −1 , I RC (s γ −1 ))y 2 (n, γ ) , s k 2 −1 γ Q 6 (s γ , A C (s γ ))y 1 (n, γ ) − s k 2 −1 γ −1 Q 6 (s γ −1 , A C (s γ −1 ))y 2 (n, γ ) , s k 2 −1 γ Q 10 (s γ , L TC (s γ ))y 1 (n, γ ) − s k 2 −1 γ −1 Q 10 (s γ −1 , L TC (s γ −1 ))y 2 (n, γ ) , www.nature.com/scientificreports/ different values of k 1 and k 2 in favour of L TC and I TC decrease to nearly zero. Finally, in Fig. 13, the graph is plotted for both disease recovered individuals against time and varying values of k 1 and k 2 . We have found that the different values of k 1 and k 2 in favour of the number of recovered people are increasing. In Figs. 14 and 15, we plotted the graph of I T and R T compared to sensitive parameters infected human individuals for different values of ρ 1 and varying k 1 and k 2 against time. Then, the number of active TB infected human individuals decreases rapidly, and recovery from TB increases with time. At the same time, we plotted the graph of I C and R C compared to sensitive parameters in reinfected human individuals for different values of ρ 2 and varying k 1 and k 2 against time in Figs. 16 and 17. Then the symptomatic infection from COVID-19 and recovery from COVID-19 first increases at the initial stages and decreases with time, it gets very close to zero. Finally, the graphs are plotted to compare all compartments against time, with the same values of k 1 and k 2 (k 1 = k 2 = 0.95) in Fig. 18. Further, in Fig. 19, we obtain the simulated results with the available real data COVID-19 infected Indians in World Health Organization from 01 st June 2022 to 08 th September 2022 for 100 days as a data case and present a  www.nature.com/scientificreports/ graphical comparison. We fixed the parameter values in these graphical results and varied the k 1 and k 2 . We see that the graphs of the simulated and real data curves are very close to each other in the final stage at the order of k 1 = k 2 = 0.92 . Our proposed model performance is good because the number of recovered people is increasing. Hence, the fractal-fractional operator is an easy tool to understand the TB and COVID-19 co-infected model.  www.nature.com/scientificreports/

Conclusions
A fractal-fractional TB and COVID-19 co-infection model is investigated in this article. Firstly, we formulated a fractal-fractional type TB and COVID-19 co-infection model to demonstrate the theoretical existence and uniqueness results under the said derivative by utilizing the fixed point approach. An examination was conducted on the criteria proposed by Ulam-Hyers stability. This paper used Lagrange polynomial interpolation to derive the numerical scheme for the TB and COVID-19 co-infection model. We can also validate the results through a numerical simulation that has been carried out for the different values for fractional order k 1 , fractal dimensions k 2 and parameters. Based on the numerical simulation, we have a graphical explanation of the model and a comparison of the sensitive parameters. The numerical portion of the paper presents highly realistic graphs for various orders of k 1 and k 2 . These comparative results exhibit similar patterns but with slight deviations corresponding to the specific orders of fractal-fractional derivatives. The numerical simulation shows that the fractal-fractional TB and COVID-19 model has performed very well, as the number of recovered people increases against time. To extend the research on the subject, we can use other numerical schemes and comparative analyses to the continuation of the study.       www.nature.com/scientificreports/

Data availability
All data regarding the research work is clearly mentioned in the research work.